Collocation approximation by deep neural ReLU networks for parametric and stochastic PDEs with lognormal inputs

Abstract

We find the convergence rates of the collocation approximation by deep ReLU neural networks of solutions to elliptic PDEs with lognormal inputs, parametrized by $\boldsymbol{y}$ in the noncompact set ${\mathbb R}^\infty$. The approximation error is measured in the norm of the Bochner space $L_2({\mathbb R}^\infty, V, \gamma)$, where $\gamma$ is the infinite tensor-product standard Gaussian probability measure on ${\mathbb R}^\infty$ and $V$ is the energy space. We also obtain similar dimension-independent results in the case when the lognormal inputs are parametrized by ${\mathbb R}^M$ of very large dimension $M$, and the approximation error is measured in the $\sqrt{g_M}$-weighted uniform norm of the Bochner space $L_\infty^{\sqrt{g}}({\mathbb R}^M, V)$, where $g_M$ is the density function of the standard Gaussian probability measure on ${\mathbb R}^M$. Bibliography: 62 titles.